I would agree with Nick here. This question doesn't really make sense given what bitcoin is.
– Jimmy SongFeb 14 '15 at 1:30

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Gold mining is not always a perfect analogy for Bitcoin mining, but in this instance it holds up pretty well. A miner digs some gold out of the ground, then goes into town on the weekend with his little bag of gold dust, and uses it to buy whiskey at the saloon. If you ask him who "issued" him the gold, expect a strange look and tobacco juice on your shoes - he dug it up for himself, you crazy city slicker. Bitcoin mining is effectively similar but without the tobacco juice.
– Nate EldredgeFeb 14 '15 at 5:17

No one entity overseas the issuance of block rewards. This is one of several revolutionary concepts behind Bitcoin. (There is absolutely no Federal Reserve.) The Bitcoin Protocol and its distributed blockchain consensus mechanism is what effectively awards miners solving a very difficult hashing puzzle. The solution block groups a number of transactions together to be added to the Blockchain, and the preponderance of other "full node" block chains must add that solution block to their records in order for it to be "recognized".

The first transaction in such a block is called the "coinbase transaction." It is paid to the miner that successfully added a block to the Blockchain. It includes the current 25 BTC award and mining transaction fees.

Solo mining is accomplished by a solo miner connecting their miner (e.g., bfgminer, cgminer) to their own bitcoin node to advertise on the Bitcoin Network a solution has been found. Similarly for mining pools, they manage one or more full nodes that advertise when the pool has discovered a solution. Mining pools ensure the multiple outputs of a coinbase transaction pays its members their dues based upon shares shares of work they have successfully completed, hopefully with fair accounting. Miners can't spend their booty until their block is 100+ blocks deep in the Blockchain.

You are responsible for issuing them. You, together with all other users.

Bitcoins are not issued, as they are not tokens that are redeemable for something else. They don't have any promised or guaranteed or even intended value. Bitcoins are simply bitcoins, and the rules of the system determine how many and who can create them.

It is however us all who are allowing that process to take place. The system exists because the result is something which is usable as money, and people choose to accept it, giving bitcoins value.

I suppose the miner does or whoever constructs the block header that's being hashed. When you construct the block, there's a special transaction called a coinbase transaction where you're allowed to assign bitcoins to yourself(or any bitcoin address of your choosing as output).

Other mines and nodes in the network will accept this transaction in the block because you also provide proof-of-work behind it.

The Bitcoin protocol defines rules which outline valid interaction in the network. One of the rules states that each Block must contain a Coinbase Transaction. The Coinbase Transaction may be used to collect the transaction fees, and to create a limited number of new coins.